On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Γ=Cay(G,S)is said to be normal if G is normal in Aut Γ.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Γ of finite nonabelian simple groups G,where the vertex stabilizer Av is soluble for A=Aut Γ and v ∈ VΓ.We prove that either Γ is normal or G=A5,A10,A54,A274,A549 or A1099.Further,11-valent symmetric nonnormal Cayley graphs of A5,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.     

Fitting Height and Diameter of Character Degree Graphs

Let a graph Γ have bounded Fitting height (i.e., there is a bound on the Fitting heights of those groups whose character degree graph is Γ) and G be any solvable group with character degree graph Γ and Fitting height h(G). We improve Moretò's bound by proving that if no vertex in Γ is adjacent to every other one, then h(G) ≤4, else h(G) ≤6. As a consequence, if a solvable group G has character degree graph with diameter 3, then h(G) ≤4. Moreover, G has at most one non-abelian normal Sylow subgroup in this case.     

A note on the girth‐doubling construction for polygonal graphs

A near‐polygonal graph is a graph Γ which has a set ?? of m‐cycles for some positive integer m such that each 2‐path of Γ is contained in exactly one cycle in ??. If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m, Archdeacon and Perkel proved the existence of a polygonal graph Γ₂ of valency r and girth 2m. We will show that this construction can be extended to one that yields a polygonal graph Γ₃ of valency r and girth 3m, but that making the cycles any longer with this construction does not yield a polygonal graph. We also show that if Aut(Γ) is 2‐arc transitive, so is Aut(Γ_k) for k = 2, 3. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 246‐254, 2011     

6p阶可解对称图的分类

设ｘ是简单无向图，Ｇ是Ａｕｔ（Ｘ）的一个于群，Ｘ称为Ｇ－对称的，如果Ｇ在ｘ的１－孤（即两相邻顶点构成的有序偶）集合上的作用是传递的；ｘ称为对称图，如果Ｘ是Ａｕｔ（ｘ）－对称的；ｘ称为可解对称的，如果Ａｕｔ（Ｘ）包含可解子群Ｇ，使Ｘ是Ｇ－对称的．本文给出了具有６Ｐ个顶点的可解对称图的一个分类，这里ｐ≥５是素数．     

Teichmuller空间的星形结构问题

对于任一保持单位圆盘Δ及其外部Δ的Ｆｕｃｈｓ群Γ，利用Ｂｅｒｓ嵌入，Ｔｅｉｃｈｍｕｌｌｅｒ空间Ｔ（Γ）可看成是Δ上Γ的有界全纯二次微分Ｂ（Δ，Γ）中的一个有界区域，本文的目的是讨论Ｔｅｉｃｈｍｕｌｌｅｒ空间Ｔ（Γ）的星形问题。特别地，我们证明了：当Γ是第二类Ｆｕｃｈｓ群时，Ｔ（Γ）不是星形的．     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

On the Commuting Graph of Dihedral Group

Let Γ be a non-abelian group and Ω ? Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D_2n. We also study the metric dimension of commuting graph on D_2n and compute its resolving polynomial.     

Mean distance in a graph

The average or mean of the distances between vertices in a connected graph Γ, μ(Γ), is a natural measure of the compactness of the graph. In this paper we compute bounds for μ(Γ) in terms of the number of vertices in Γ and the diameter of Γ. We prove a formula for computing μ(Γ) when Γ is a tree which is particularly useful when Γ has a high degree of symmetry. Finally, we present algorithms for μ(Γ) which are amenable to computer implementation.     

A class of symmetric graphs with 2‐arc transitive quotients

Let Γ be an X‐symmetric graph admitting an X‐invariant partition ?? on V(Γ) such that Γ_?? is connected and (X, 2)‐arc transitive. A characterization of (Γ, X, ??) was given in [S. Zhou Eur J Comb 23 (2002), 741–760] for the case where |B|>|Γ(C)∩B|=2 for an arc (B, C) of Γ_??.We con‐sider in this article the case where |B|>|Γ(C)∩B|=3, and prove that Γ can be constructed from a 2‐arc transitive graph of valency 4 or 7 unless its connected components are isomorphic to 3 K ₂, C ₆ or K _{3, 3}. As a byproduct, we prove that each connected tetravalent (X, 2)‐transitive graph is either the complete graph K ₅ or a near n‐gonal graph for some n?4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 232–245, 2010     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

Isomorphisms of finite semi-Cayley graphs

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph(Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph(also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits(of equal size). In this paper, we introduce the concept of SCI-graph(semi-Cayley isomorphism)and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.     

Simple Minimum (K4-e)-coverings of Complete Multipartite Graphs

A decomposition of K_(n(g))∪Γ, the complete n-partite equipartite graph over gn vertices union a graph Γ(called the excess) that is a subgraph of K_(n(g)), into edge disjoint copies of a graph G is called a simple minimum group divisible covering of type g~n with G if Γ contains as few edges as possible. We examine all possible excesses for simple minimum group divisible(K_4-e)-coverings.Necessary and sufficient conditions are established for their existence.     

Uniqueness of conformal densities and the semiflow of Weyl chambers

Let X = G/K be a symmetric space of noncompact type and Γ a discrete “generic” subgroup of G with critical exponent δ(Γ). We show that, if Γ is of divergence type, then there is a unique Γ-invariant conformal density of dimension δ (Γ) (hence a Patterson-Sullivan density) on the set of regular elements of the geometric boundary of X. This problem is directly related to the recurrence of the semiflow of Weyl chambers on X/gG.     

本原(J1,2)-弧传递图的分类

In the author's Ph. D thesis, a non-quasiprimitive graph admitting a quasiprimitive automorphism group isomorphic to J1 was constructed,where J1 is Janko simple group of order 175560.Is this the only one for J1?In this paper all primitive(J1,2)-arc transitive graphs Γ are given and that AutΓ≌J1 is proved.     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

p~2阶弧传递循环图的正规性条件

群G关于S的有向Cayley图X=Cay(G,S)称为pk阶有向循环图,若G是pk阶循环群.利用有限群论和图论的较深刻的结果,对p2阶弧传递(有向)循环图的正规性条件进行了讨论,证明了任一p2阶弧传递(有向)循环图是正规的当且仅当(|Aut(G,S)|,p)=1.     

THE NORMALITY OF CAYLEY GRAPHS OF SIMPLE GROUP A5 WITH SMALL VALENCIES

Let G be a finite group and S a subset of G not containing the identity element 1. We define the Cayley (di)graph X = Cay(G, S) of G with respect to S by V(X) = G,E(X) = {(g, sg) [ g ∈ G, s ∈ S}. A Cayley (di)graph X = Cay(G, S) is called normal if GR A = Aut(X). In this paper we prove that if S = {a, b, c} is a 3-generating subset of G = A5 not containing the identity 1, then X = Cay(G, S) is a normal Cayley digraph.     

Overlap in consistent cycles

A (directed) cycle C in a graph Γ is called consistent provided there exists an automorphism of Γ, acting as a 1‐step rotation of C. A beautiful but not well‐known result of J.H. Conway states that if Γ is arc‐transitive and has valence d, then there are precisely d ? 1 orbits of consistent cycles under the action of Aut(Γ). In this paper, we extend the definition of consistent cycles to those which admit a k‐step rotation, and call them ‐consistent. We investigate ‐consistent cycles in view of their overlap. This provides a simple proof of the original Conway's theorem, as well as a generalization to orbits of ‐consistent cycles. A set of illuminating examples are provided. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 55–71, 2007     

Commutativity Pattern of Finite Non-Abelian p-Groups Determine Their Orders

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ _G (called noncommuting graph of G) with G as follows: Take G?Z(G) as the vertices of Γ _G , and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that “the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that Γ _P  ? Γ _H for some group H, then |P| = |H|.     

Steinberg triality groups acting on 2 - (v, k, 1) designs

A 2 - (v,k,1) design D = (P, B) is a system consisting of a finite set P of v points and a collection B of k-subsets of P, called blocks, such that each 2-subset of P is contained in precisely one block. Let G be an automorphism group of a 2- (v,k,1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ≤ G ≤ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to 3D4(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.